If $(X,d)$ is a metric space and $F$ is a nonnegative lower semi-continuous function on $X$, then $F$ can be written as the sup of increasing sequence of uniformly continuous functions:
$F_n(x) = \inf_{y\in X} \{F(y) + n d(x,y)\}$.
It's clear to me that $F_n$ is continuous and pointwisely converges to $F$. I'm wondering why $F_n$ is uniformly continuous on $X$?
Any hint will be appreciated.
